An incompressible viscous liquid occupies the long thin region $0 \leqslant y \leqslant h(x)$ for $0 \leqslant x \leqslant \ell$, where $h(x)=d_{1}+\alpha x$ with $h(0)=d_{1}, h(\ell)=d_{2}<d_{1}$ and $d_{1} \ll \ell$. The top boundary at $y=h(x)$ is rigid and stationary. The bottom boundary at $y=0$ is rigid and moving at velocity $(U, 0,0)$. Fluid can move in and out of the ends $x=0$ and $x=\ell$, where the pressure is the same, namely $p_{0}$.

Explaining the approximations of lubrication theory as you use them, find the velocity profile in the long thin region, and show that the volume flux $Q$ (per unit width in the $z$-direction) is

$$Q=\frac{U d_{1} d_{2}}{d_{1}+d_{2}}$$

Find also the value of $h(x)$ (i) where the pressure is maximum, (ii) where the tangential viscous stress on the bottom $y=0$ vanishes, and (iii) where the tangential viscous stress on the top $y=h(x)$ vanishes.